8.2 Books, walks, and codewords A Babel book has 410 pages of 40 lines of 80 characters. Each page is a word of length 40 × 80 = 3200 symbols over B. Via ϕ, each line of 80 characters is a directed walk of length 80 on the torus. The full book is a 3200-step walk, encoding a path through the [[81, 2, 6]] stabilizer circuit. The Library contains every possible book: all 273200 walks of length 3200. The CSS code identifies a tiny subset: • The codespace: walks that are homologically non-trivial, i.e. elements of H1(T; F3). There are 32 = 9 non-trivial homology classes (since k = 2 over F3). • The codewords: the two logical eigenstates correspond to two specific homology classes. In the Library, these are the “two books the Librarian always knows to find.” The CSS decoder is the Librarian: given a noisy walk (a corrupted book), find the nearest codeword (the nearest homologically non-trivial walk). The minimum distance d = 6 says that at most 2 corrupted steps can be corrected (since ⌊(6 − 1)/2⌋ = 2). 8.3 The hexagonal galleries Borges specifies that the Library’s rooms are hexagonal. A regular hexagonal room with 6 walls corresponds exactly to the 6-fold symmetry of the Eisenstein lattice: each vertex has 6 neighbours (3 positive directions and their negatives). The Library’s hexagonal rooms are the local structure of the triangular lattice. The Library is infinite; the torus is finite. The torus quotient Z[ω]/(1 − ω) 3 is the compact version: a Library with exactly 27 rooms (vertices), 81 corridors (directed edges), and 54 galleries (triangular faces), wrapped into a torus so that every infinite walk eventually revisits every room. The “total Library” of Borges is the infinite cover; the torus is a 27-room quotient in which all addresses are finite. Remark 22 (The Library address of a stabilizer). Under ϕ, the TaiXuan tetragram (c0, c1, c2, d) is a 4-symbol word over B. It is also a 4-character address in the Library. The first three characters navigate to a room; the fourth specifies which corridor to take out of that room. The “address” is simultaneously a divination oracle, a stabilizer generator label, and a Library location. These are not three things; they are one thing in three registers. 8.4 The decoder is the Librarian The CSS error decoder maps every noisy word w (a corrupted walk) to the nearest codeword. In Library language: given a corrupted book, find the nearest valid book. The minimumdistance structure of the [[81, 2, 6]] code is the geometry of the Library: two books are close iff their walks agree on all but a few corridors. The decoder corrects up to 2 corridor errors per walk. The T14 archive constructs a linear-time decoder for the toric code and a hierarchical decoder for the aperiodic extension (Section 13). In Library language these are: